Mediumship

Several characters have communicated with the dead. Hurley sees, talks with and even plays chess with dead friends. The Man in Black as a child saw his mother's ghost. Young Ben saw his dead mother on the island as well.

Miles is a professional medium, but he says he doesn't talk to the dead. Instead, he senses the dead's last thoughts.

Premonitions

Several characters have foreseen future events. These premonitions usually come in the form of dreams. Desmond, however, foresaw events as specific flashes.

Psychic intuition

Miles exhibited some psychic ability independent from his main ability. He intuited Michael's true identity on the Kahana, and he knew Charlotte had been searching for the island, knowledge which surprised her. He also suggested he knew other freighter passengers' secrets.

Walt displayed supernatural knowledge of the island. He once told Locke "Don't open it, don't open that thing," despite never having seen or heard of the Hatch. He showed further more mysterious behavior that night, suddenly telling his father without explanation that they now had to leave the Island. Off-island, Walt dreamed of Locke, so wasn't surprised when he came to visit. ("Born to Run") ("The Life and Death of Jeremy Bentham")

The Man in Black, who was supposedly sent into the Heart of Island, came out as black smoke due to the lack of his resistance to the electromagnetism. He was still "alive" as a smoky form, which might be the effect of the immortality bestowed upon him. His immortality made him take a smoky form (soul), which could assume the appearance of any other dead person.

Physical resistance

The Man in Black, in both human and monster forms, resisted all physical attacks, including knives and bullets. He lost this resistance when Desmond turned off the source of his powers.

Shapeshifting

Summoning animals

An Australian Bronze cuckoo once flew to its death into Walt's home's glass door. Walt was reading about Australian Bronze cuckoos in a text book at the time. On the island, when the Others imprisoned him in Room 23, several birds flew into a nearby boarded window, again to their deaths. The Others realized they'd got "more than they bargained for" with Walt, and they returned him to Michael. ("Room 23") ("Special") ("Live Together, Die Alone, Part 1")

Telekinesis

The Man in Black freed Ben from his shackles without physically touching them. Shortly before his death, his boat arrived on the main island, though it had last been docked on Hydra Island. ("Dr. Linus") ("The End")

Teleportation and astral projection

Walt has more than once, as Bea Klugh put it, appeared "somewhere he wasn't supposed to be".
Not long after the Otherskidnapped him, Walt appeared in front of Shannon, soaking wet and whispering incomprehensibly. Three nights later, he appeared to her again, and Sayid saw him as well the following day. Seeing Walt led Shannon to her death. A taller, older Walt later appeared to Locke, stopping him from killing himself. The show has not revealed whether these appearances were Walt himself, visions of manifestations of the Man in Black. ("Adrift") ("Abandoned")